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An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution $u=\tan (x / 2)$ or, equivalently, $x=2 \tan ^{-1} u .$ The following relations are used in making this change of variables.$$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$Verify relation $A$ by differentiating $x=2 \tan ^{-1} u .$ Verify relations $B$ and $C$ using a right-triangle diagram and the double-angle formulas$$\sin x=2 \sin \frac{x}{2} \cos \frac{x}{2} \quad \text { and } \quad \cos x=2 \cos ^{2} \frac{x}{2}-1$$.

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

Integration Techniques

Section 5

Partial Fractions

Integrals

Integration

Applications of Integration

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okay. Trying to verify some relations. Starting with X equals two times the inverse tangent of you. Well, that means if we solve this for you, me to have X over two equals are inverse tangent of you. So that means you would equal the tangent X over two. And what they want to do is figure out what DX is first. So the derivative of X is gonna be the derivative of our right inside. So that would be two times derivative up. Inverse tangent of an angle function is one over the ankle function squared plus one. Did you? So that means that d x there's gonna be two over you squared possible in, do you? So that's first identity that they wanted us to verify. And then they wanted this to verify a couple of others using our right triangle relationships. So if we have the angle of X over two and if we want the tension of that angle of tangent is the opposite leg over the adjacent leg. So if that equals you and that means the opposite leg must be you and the adjacent leg must be one. And then from the Pythagorean Theorem the pot news is going to be the square root off one plus u squared. So now we know that the sign five x over two is gonna be the opposite Over the high part news you over the square root of one plus u squared and the coal sign of X over two is gonna be the adjacent leg, which is one over the iPod news, the square root of one plus u squared and first thing we want us to verify waas The sign of X well from our trig ratios, our relationships with the different trig ratios and 1/2 angle and double angle formulas. The sign off X is two times the sign of half of X, which would be you over our square root of one plus u squared times the co sign of half of X, which would be one over the square root one plus u squared and then just multiplying that all together we end up with sign of X, it's gonna be to you over one plus u squared and then the co sign of X. We got a couple of different formulas for that one. I'm gonna use the two times Geico sine squared of X over two buying this one. So that would be two times my one over square root of one plus U squared squared minus one. So that will give me to over one plus U squared minus to turn that one into the same denominator. One plus U squared over one plus u squared, and now I can combine my like terms. Tu minus one is to have started one tu minus one is one and then minus the U squared over one plus u squared.

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